Theoretical calculation of the relative abundance between He- and Li-like Ti under the coronal condition

Journal of Quantitative Spectroscopy & Radiative Transfer 74 (2002) 511 – 519

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Theoretical calculation of the relative abundance between He- and Li-like Ti under the coronal condition Xiang-dong Li ∗ , Shen-sheng Han, Cheng Wang, Zhi-zhan Xu Laboratory for High Intensity Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, People’s Republic of China Received 18 June 2001; accepted 5 October 2001

Abstract This paper provides a theoretical calculation of the relative abundance between He- and Li-like Ti in the electronic temperature range from 0:1 to 20 keV under the coronal conditions. The results show that the abundance of Li-like Ti is ¡ 6 percent of He-like Ti as the electronic temperature exceeds 3 keV, and this percentage will further decrease when the photon-ionization is considered. Thus, it is proven that for the Ti plasma it is approximately true that there are only two charge states (H- and He-like) in the plasma when the electronic temperature is beyond 3 keV. The results also show that the abundance of Li-like Ti will reach a maximum in temperatures range from 0.6 to 0:7 keV. In this paper, the Younger formula is used to estimate the electron–ion collision rates and it is shown that this formula is in a good agreement with experiment data and is better than other theoretical methods. Dielectronic recombination of He-like Ti is considered by a full quantum mechanics method and the total electronic recombination rates of He-like Ti will reach the maximum values at the temperature 2:9 keV, which is close to the plasma temperature of 3 keV formed as in Shepard et al. (Phys. Rev. E 53 (1996) 5291). ? 2002 Elsevier Science Ltd. All rights reserved. PACS: 52.70.−m; 52.25.Dg; 34.80.Kw; 34.80.Lx Keywords: Relative abundance; He-like Ti; Li-like Ti; Coronal condition

1. Introduction The relative abundance of diBerent charge states is one of the important plasma’s parameters, which can be used in studies on plasma properties, such as in diagnosis of the electronic temperature by the line ratio [1–3]. The relative abundance of diBerent charge states is indispensable in calculating the relative intensity of diBerent spectrum lines from diBerent ion states. Recent studies on diagnostics of ∗

Corresponding author. Tel.: +86-21-5953-2254; fax: +86-21-5952-8812. E-mail address: xiangdong [email protected] (X.-d. Li).

0022-4073/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 1 ) 0 0 2 5 0 - 3

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electronic temperature, which involved both inter-stage and isoelectronic line ratios, all depend on the relative abundance of diBerent charge states [4]. In Shepard’s recent work [1], in which an analytic method is used to compare with the temperature diagnostic results from Ration [5] in laser produced plasma, Ti and Cr were chosen as the tracers. H- and He-like charge states were then supposed to be the only two charge states for Ti and Cr in the plasma. It is evident that this supposition cannot be true for all electronic temperature and thus it is necessary to calculate the relative abundance between He- and Li-like charge states so that we can Jnd out at what temperature the abundance of Li-like charge state become unimportant compared with the He-like charge state. Just as in the Shepard’s work, to simplify the calculations we have also chosen the coronal model in this paper. It will be seen that the results are still useful to the laser-produced long-scale plasma (such as in inertia conJnement fusion ICF) for analytical purpose. Calculation of the relative abundance of diBerent charge states depends on solving a series of coupled rate equations and calculating a variety of rate coeLcients. Generally, these rate equations cannot be solved easily and the rate coeLcients may also not be well established. The usual way of solving such rate equations is through a proper modeling for speciJc plasma, i.e. to construct a relative simple model for the plasma so that the main dynamic process in such kind of plasma can be considered. So far, several basic plasma models have been constructed, for example, the coronal model, the nebula model and the atmosphere model etc. In this paper, coronal model is chosen so as to simplify the problem. A full quantum mechanics calculation for the rate coeLcients is diLcult, sometimes even impossible, and thus when the rate coeLcients are needed for the diagnostic purpose, the semi-classical or empirical methods need to be employed to approximate the calculation of the rate coeLcients and again several methods can be used. In Younger [6] a formula was used to calculate the electron–ion collision ionization rates for the purpose of plasma modeling, which proved to be a good method. Dielectronic recombination is another important process in the coronal plasma. The eBects of which in the high temperature region is still an interesting Jeld. Most diagnostic work uses the Burgess formula [7,8] to calculate the dielectronic recombination rates. We have found, however, that the results diBer greatly between the calculations from Burgess’ formula and full quantum mechanics, especially for the non-hydrogen-like ions. Thus, in order to accurately test the eBects of dielectronic recombination as the electronic temperature increases, we have used detailed quantum mechanics calculations for dielectronic recombination rates. The results show that in the coronal plasma the eBects of dielectronic recombination are greater than those of spontaneous radiative recombination as the electronic temperature increases and thus, in the high temperature region the dielectronic recombination will greatly aBect the relative abundance. At 2:9 keV the dielectronic recombination rates reach a maximum and this temperature is just close to the plasma temperature produced in Hohlraum by Nova laser mentioned in Ref. [1]. 2. Theory and analytic treatment As the coronal model has been used in this paper, the main dynamic processes in plasma should include the electron–ion collision ionization, the spontaneous radiative recombination and the dielectronic recombination (usually the dielectronic recombination takes eBects in the high temperature region). Another important property in the coronal plasma is that the charge states are almost all in

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their ground state. Thus, the initial ion states of the recombination and ionization processes are only treated by taking the ground states into account. This can greatly simplify the calculation process. The following sections detail our treatments for the three processes mentioned earlier. 2.1. Electron–ion collision ionization In calculation of the electron–ion collision ionization rates we use the Younger formula [6], in which electron–ion collision ionization cross section are Jtted according to both experimental results and results calculated by DWE (distorted wave exchange approximation). In order to calculate an ionization rate from a cross section, it is necessary to know the cross section at some electron energy, and in order to cover fully the distribution of electron velocity, the electron energy must cover at least two or three times of electron main thermal energy. A substantial reduction in the eBorts involved in calculating cross sections over an extended energy region may be accomplished by noting that at very high energy the ionization cross section tends to the Bethe limit or 1 lim Q = C ln(u); u

u→∞

(1)

where u is the incident electron energy in ionization threshold units and u = E=I;

(2)

where I is the ionization energy and C is the Bethe constant. By Jtting the low energy cross-section data (from DWE and experiments) and asymptotic Bethe limit, Younger found the parameterized scaled ionization cross section to be 

1 2 1 D 2 uI Q = A 1 − +B 1− (3) + C ln u + ln u: u u u After integrating over a Maxwellain electron velocity distribution, the electron impact ionization rate can be obtained as 
3=2   ∞ I 3=2 7 I S = 6:70 × 10 (uI 2 Q) exp(−uI=kT ) du; (4) kT 1 where I is ionization energy in eV, S the rate coeLcient in cm3 =s, and uI 2 Q is given by Eq. (3). The parameters A is of the form A=

3  n=0

an

1 ; (Z − N + 1)n

(5)

where Z is the nuclear charge, N the number of electrons in the initial ion and an are coeLcients. Similar expressions occur for B, C and D. The coeLcients an of these isoelectronic Jts are given in Ref. [6], in which an can be found from H- to Ne-like ions. The total electron–ion collision ionization rates of Li-like Ti in the ground state are calculated.

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2.2. Dielectronic recombination The total dielectronic recombination rates of He-like Ti have been calculated using full quantum mechanics method. The state-to-state dielectronic recombination rates can be written as [9]
 Aajm0 Arjk 43=2 a30 −E jm0 =T gj d d   r jk (T ) = m0 jk (T ) = e (6) a T 3=2 gm 0 m
Ajm
+ k
Ajk
where m0 expresses ground state of the He-like Ti, j and k are autoionization state and nonautoionization excited state of Li-like Ti, respectively. The symbols gj and gm are statistic weights for a corresponding state. Aajm0 is autoionization coeLcients, where    a Aji ; (7) Ajm0 = i

Jxed m0 and j

Aa

where ji = 2=˜|j|H |i|2 . The symbol i is a state like the ground state of He-like Ti plus a free electron. Aaji are calculated by Hartree–Fock method. The total dielectronic recombination rates are  d jk ; (8) d (T ) = j

k

which is a summation over j and k where j and k are both inJnite. Details related to the choice of j and k are given in Ref. [10]. 2.3. Spontaneous radiative recombination Spontaneous radiative recombination is from Spitzer [11] (and equivalent to the treatment of Seaton [12] with the Gaunt factor being equal to 1): √ 26  2 r (9) a = 3=2 re cZU 3=2 exp(U )E1 (U ) 3 where ar is the spontaneous radiative recombination rate for a particular initial ionization stage, re is the classical electron radius, c is the speed of light and Z is the charge on the initial ionization stage, including inner-electron screening. E1 (U ) is the exponential integral of Jrst kind [13]. As in the Ration model the Eq. (9) is also used in this paper for recombination to a particular state. In electron–ion collision ionization only the ground state are taken into account, because the populations of the excited states in impacted ion are very small compared with the populations of the ground state in impacted ion under coronal conditions. With a recombination process the recombination rates are related with the populations of the levels of the recombining ion, while there are no relations with the populations of the levels of the recombined ion. In the case of the recombination process, since the coronal conditions are considered, only ground state is taken into account for the recombining ion. Thus, for the total spontaneous radiative recombination rate, Elwert’s [13] approach is used. The total spontaneous radiative recombination rate, which consider the recombination to excited states, can be written as, ar = 2:06 × 10−11 (G=m)G1 (m =kT )Ze2 T −1=2 ; 1:36

0:43

(10) y

where G ≈ 1 + 0:37m + 0:25m (m =kT ) ; G1 (y) = ye E1 (y), E1 (y) is the exponential integral of Jrst kind. ze is the eBective nuclear charge. m = EH Ze2 =m2 is the energy of ionization of the

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recombined ion in the ground state with the principal quantum number m. In this paper the total spontaneous radiative recombination rates of He-like Ti to Li-like Ti are calculated. 2.4. Ionization equilibrium The steady-state ionization balance equation in this case is Ne NHe adHe→Li + Ne NHe arHe→Li = Ne NLi SLi→He ;

(11)

where N are the particle densities for the corresponding index. So the relative abundance between He-like and Li-like Ti can be obtained as NHe SLi→He = d : NLi aHe→Li + arHe→Li

(12)

For laser-produced long-scale plasma the equilibrium can also be achieved through electron collision process (as in Ref. [1]). So the Eqs. (11) and (12) will also hold for long-scale plasma after adding the photon-ionization process detailed in Eqs. (11) and (12). If the photon-ionization process is included the values of NHe =NLi will increase. According to the methods above, we can make an entire program to calculate the diBerent rates and the relative abundance. The results and discussion are given in the Section 3. 3. Results and discussion Fig. 1 shows the electron–ion collision ionization rates of one 2s electron and two 1s electrons and the total ionization rates for Li-like Ti in its ground state. All the results are from Younger

Ionization rate coefficients(cm3sec-1)

4.00E-011 3.50E-011

rates for one 2s electron ionization rates for two 1s electron ionization total rates

3.00E-011 2.50E-011 2.00E-011 1.50E-011 1.00E-011 5.00E-012 0.00E+000 -5.00E-012 0

5

10

15

20

Te (keV)

Fig. 1. Results of the collision ionization rates for Li-like Ti in the ground state. All the results are calculated by Younger formula. The rates for 1s ionization including two electrons.

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Table 1 Electron-impact ionization cross section of Li-like Ti in units of 10−21 cm2

Cross section

Ti19+

Ea (keV)

Gea

vHa

DWa

Lotza

Average

Younger

3.33

6:67 ± 1:33

6:98 ± 0:75

7.81

8.04

6:91 ± 0:65

7.185

a

E is the incident electron energy. Ge and vH represent measurement from diBerent experiments [14]. Average represents the weighted average of the Ge and vH results. “DW” and “Lotz” are the distorted wave and Lotz [15] formula results, respectively, Zhang and Sampson [16]. 3.50E-012 3.00E-012

rates (cm3sec-1)

2.50E-012 2.00E-012 1.50E-012 1.00E-012 5.00E-013 0.00E+000

Total spontenous radiative recombination rates Total dielectronic recombination rates

-5.00E-013 -1.00E-012 0

5

10

15

20

Te (keV)

Fig. 2. The spontaneous radiative recombination is from He-like Ti to Li-like Ti. The He-like Ti is considered only in the ground state, but the recombination to Li-like excited states are considered. The dielectronic recombination is also from He-like Ti to Li-like Ti. He-like Ti is also in the ground state.

Jtted formula. The total ionization rates are the summation of the 2s ionization and 1s ionization rates. From Fig. 1 we Jnd that in high temperature region the contributions of the two 1s electrons to the total ionization rates increase more with electron temperature than for 2s electron does. Thus, in the high temperature region the inner shell ionization had to be considered in the plasma rate equation, while in low temperature region the inner shell ionization can be ignored. In order to check the accuracy of Younger formula, we compare the collision ionization cross section between experiments in Ref. [14] and theoretical results obtained at an incident electron energy of 3:33 keV detailed in Table 1. It is clear that the Younger formula gives better results than both Lots and DW methods and thus the Younger formula is a good empirical method that can be used in the plasma modeling. Fig. 2 illustrates the total rates of spontaneous radiative recombination and dielectronic recombination from He-like Ti to Li-like Ti. It can be seen in Fig. 2 that the spontaneous radiative

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with dielectronic recombination without dielectronic recombination results from Seaton formula

400

NHe/NLi

300

200

100

0 0

5

10

15

20

Te(keV)

Fig. 3. Comparison of the relative abundance NHe =NLi with and without dielectronic recombination. The NHe =NLi results from Seaton formula is also provided.

recombination rates decrease rapidly in the low temperature region and change smoothly in hightemperature region. In the high-temperature region the tendencies of dielectronic recombination rates and spontaneous radiative recombination rates become similar and their quantities are of the same order. Thus, on increasing the temperature dielectronic recombination become relative important. From Fig. 2 we can also see that at 2:9 keV the dielectronic recombination rates reach a maximum at a temperature near the plasma temperature (about 3 keV) produced in Hohlraum by Nova laser in Ref. [1]. This means that as in the plasma formed in Ref. [1] dielectronic recombination get the best results. According to these results, the relative abundance between He- and Li-like Ti are calculated by Eq. (12) in the temperature range from 0.1 to 20 keV, with the results shown in Fig. 3. Fig. 3 shows that the relative abundance NHe =NLi increase with the electron temperature and this is physically true. We Jnd from our results, which considered the dielectronic recombination, that as the temperature change from 0.6 to 0:7 keV the values of NHe =NLi change from less than one to greater than one. We can thus deduce that in this temperature interval the Li-like charge state of Ti can reach its maximum distribution. This means that when the temperature is greater than 0:7 keV the abundance of the charge states, for which the degree of ionization is less than that of Li-like ion, are less than the abundance of Li-like charge state. In Fig. 3, we also provided the results of NHe =NLi from Seaton formula, which is the formula to calculate the relative abundance of two neighbor charge states under coronal conditions. The results from Seaton are close to our results calculated without dielectronic recombination. But there are some diBerences. The results from Seaton are so diBerent from our results with the dielectronic recombination and the discrepancies between them increase as the temperature increase. The estimations of NHe =NLi from Seaton are all greater than the estimations in this paper. The Seaton formula may consider relative more ionization, so that the temperature, at which certain charge state will reach its maximum distribution, is lower than the estimation in our calculation. From Seaton formula NHe =NLi = 1 appears at Te = 0:444 keV, while it is at between 0.6 to 0:7 keV in our results.

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In almost all the temperature diagnostics using the line ratio, the abundance or relative abundance of H- and He-like charge states are used. Particularly, in the temperature diagnostics using the isoelectronic line ratio, the abundance of all charge states must be known. This is diLcult to do. But if the abundance of other charge states can be ignored on comparison with H-like and He-like charge states, the problem will be greatly simpliJed. In Fig. 3, the relative abundance NHe =NLi increase with the temperature, so with the temperature excess at certain values the abundance of Li-like ion is unimportant compared with the He-like ion. At this juncture the plasma can be considered to be constructed of just H-like and He-like charge states. It was found in Ref. [1] that the plasma temperature, which is formed in Hohlraum by Nova laser, is at least 3 keV. At this temperature the values of NHe =NLi are about 16 (66 from Seaton). That means NLi are 6 percent of NHe and this percentage will decrease when photon-ionization is included. So the analytic treatment for Ti plasma in Ref. [1] are only reasonable at high temperature. In the low temperature region the analytic treatment in Ref. [1] may not be true. In Fig. 3 we also compare the values of NHe =NLi with and without dielectronic recombination. The diBerences are shown clearly in high-temperature region. With the increase of temperature, the eBects of dielectronic recombination become greater than the eBects of spontaneous radiative recombination. As there are so much data for NHe =NLi (with dielectronic recombination), so we Jtted an analytic form for NHe =NLi as follows:

(0:4 keV 6 T 6 1 keV; errors 6 0:03); 0:21899 − 2:47785T + 6:0758T 2 − 0:63636T 3 NHe = NLi −1:32571 + 5:03534T + 0:23233T 2 − 0:00357T 3 (1:1 keV 6 T 6 20 keV; errors 6 0:5); where T is the electron temperature. The errors were compared with results from Eq. (12). Acknowledgements This work was supported by National High Technology Program of China. References [1] Shepard TD, Back CA, Kalanter DH, KauBman RL, Keane CJ, Klem KE, Lasinski BF, MacGowan BJ, Powers LV, Suter LJ, Turner RE, Failor BH, Hsing WW. Phys Rev 1996;E53:5291. [2] Back CA, Kalantar DH, KauBman RL, Lee RW, MacGowan BJ, Montgonery DS, Powers LV, Shepard TD, Stone GF, Suter LJ. Phys Rev Lett 1996;77:4350. [3] Marjoribanks RS, Richardson MC, Jaanimagi PA, Epstein R. Phys Rev 1992;A 46:1747. [4] Abdallah Jr J, Clark REH, Keane CJ, Shepard TD, Suter LJ. JQSRT 1993;50:91. [5] Lee RW, Whitten BL, Stout RE. JQSRT 1984; 32: 91; Lee RW. User manual for ration. Livermore: Lawrence Livermore National Laboratory, 1990. [6] Younger SM. Phys Rev 1980;A 22:1425; Younger SM. Phys Rev 1980;A 22:111; Younger SM. Phys Rev 1981;A 23:1138; Younger SM. Phys Rev 1981;A 24:1272; Younger SM. Phys Rev 1981;A 24:1278; Younger SM. JQSRT 1981;26:329; Younger SM. JQSRT 1982;27:541.

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